Navier–Stokes equations
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Derivation of Navier-Stokes Equations
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Today, we'll delve into what the Navier-Stokes equations are. They describe the motion of viscous fluids. So, who can tell me what we mean by a 'viscous fluid'?
A viscous fluid is one that has a certain resistance to flow, like honey.
Exactly! Viscous fluids don't flow as easily as, say, water. The Navier-Stokes equations account for this resistance. Let's also discuss the term 'deformation law.' What might it refer to?
Is it about how fluids change shape when they flow?
Yes! The deformation law links strain rates to stress in a fluid. It’s fundamental in deriving the Navier-Stokes equations. Remember, strain rates can be represented using the acronym D.O.V. for Deformation-Orientation-Velocity.
So, is the Navier-Stokes equation one single equation?
Great question! The Navier-Stokes equations are actually a set of equations. They stem from applying Newton's second law to fluid motion. Let's summarize: Navier-Stokes = Newton’s Law + deformation law. This relationship is key!
Mechanical vs. Thermodynamic Pressure
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Now, let's talk about mechanical and thermodynamic pressures. Does anyone know how they differ?
I think thermodynamic pressure relates to temperature changes while mechanical pressure is about forces on the fluid?
Exactly! Mechanical pressure is influenced by deforming stresses, while thermodynamic pressure reflects the state of the fluid. A quick way to remember is T.M=Temperature Mechanics. Pressure equivalency occurs when the divergence of velocity equals zero. Can anyone say why that’s important?
Because we often assume incompressible flow in fluids like water!
Right! The assumptions of incompressibility simplify calculations and lead to practical applications. And this brings us to the Stokes Hypothesis. Who can summarize its importance?
It indicates conditions under which our assumptions about pressure hold true, especially in compressible fluids.
Spot on!
Incompressible and Inviscid Flow
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Now let's consider specific cases of fluid flow: incompressible and inviscid. Who remembers what 'incompressible' means?
It means the fluid density remains constant, right?
Good job! When a fluid is incompressible, the divergence of velocity becomes a key term that simplifies the Navier-Stokes equations. What would happen if we assume viscosity is negligible?
Then we transition to the Euler equations!
Exactly! The Euler equations are fundamental for understanding inviscid flow. You can also integrate Euler’s equation to derive Bernoulli's equation. Who here can explain Bernoulli's equation in simple terms?
It's about conserving energy in flowing fluids, right?
Excellent! Energy conservation is crucial in fluid mechanics. Remember, Navier-Stokes yields a broad spectrum of fluid behavior understanding!
Applications of Navier-Stokes Equations
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To wrap up, let’s connect the dots between these equations and their real-world applications. Who can think of an example of where these equations might be used in engineering?
Perhaps in designing hydraulic systems?
Absolutely! Hydraulic systems rely on principles embedded in the Navier-Stokes equations. What about in environmental studies?
They can be used to model pollutants in bodies of water!
Exactly! They help model fluid behavior across various fields. Remember, the breadth of application makes Navier-Stokes equations fundamental in fluid dynamics!
Introduction & Overview
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Quick Overview
Standard
This section delves into the derivation and significance of the Navier-Stokes equations for Newtonian viscous fluids. It highlights the distinction between thermodynamic and mechanical pressures, the implications of the Stokes Hypothesis, and further modifications for incompressible and inviscid flows, underscoring their relevance in hydraulic engineering.
Detailed
The Navier-Stokes equations are fundamental to the study of fluid mechanics, particularly in hydraulic engineering. They are derived from the general deformation law for Newtonian viscous fluids and describe the conservation of momentum in fluid dynamics. This section differentiates between thermodynamic and mechanical pressure, introducing the concept that for incompressible flow, the two pressures can be considered equivalent under certain conditions, notably when the divergence of velocity is zero or when specific assumptions about the fluid properties are made. The Stokes Hypothesis is discussed, which provides conditions under which the mechanical stress contributions reduce the equations' complexity. Furthermore, the section explains how the Navier-Stokes equations transform when dealing with incompressible flows, highlighting practical applications in hydraulics. It also introduces Euler's equation for inviscid flow and briefly discusses its integration into Bernoulli's equation, thus framing the foundational aspects of fluid dynamics necessary for further studies in computational fluid dynamics.
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Introduction to Navier–Stokes Equations
Chapter 1 of 4
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So now going back to the objective of this module the Navier–Stokes equations okay. So desired momentum equation for general Newtonian viscous fluid is obtained by equation 18 that is the deformation law in rewritten Newton’s Law which was equation number 9 and the result is famous equation of motion called as Navier–Stokes equation so it is This is equation number 20 and now we have written the general Navier–Stokes equations.
Detailed Explanation
The Navier-Stokes equations describe how fluid substances behave under various forces. They are derived from the fundamental principles of fluid dynamics, particularly from Newton's laws of motion. In this case, the equations we refer to are adapted specifically for Newtonian fluids, which have a linear relationship between stress and strain rate. By rewriting Newton's laws using the previous deformation law, represented by equation 18, we arrive at the Navier-Stokes equations, which are essential for modeling fluid flow. These equations are complex, but they help predict how fluids move in different scenarios.
Examples & Analogies
Imagine a river flowing. To understand how water moves around rocks, bends, and changes speed throughout the river, scientists need a detailed set of rules. The Navier-Stokes equations act like a recipe that tells us how the water (the fluid) will react when it encounters obstacles or changes in elevation, similar to how a recipe tells you the steps to bake a cake.
Incompressible Flow and Navier-Stokes Equations
Chapter 2 of 4
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So to do that we go to a fresh page so incompressible flow what is that if rho is that means incompressible flow that means divergence of V = 0. So if you look at the Navier–Stokes equations above this will be 0 this the last term divergence of V all right. Therefore, in equation 20 if we assume mu is constant okay as well because in most of the hydraulic purposes we assumemu is constant. Then we get Navier–Stokes equations for that is This is another equation of significance because when it comes to Navier–Stokes equations for practical purposes this is the equation that we are going to use.
Detailed Explanation
In fluid dynamics, 'incompressible flow' refers to a situation where the density of the fluid (denoted as rho) remains constant. When this condition is met, the term 'divergence of V', which describes how the fluid's velocity field spreads out or compresses, becomes zero. This assumption simplifies the Navier-Stokes equations significantly. If we also assume that viscosity (represented as mu) is constant, the resulting form of the Navier-Stokes equations becomes more practical for everyday applications like water flow in pipes or rivers.
Examples & Analogies
Think of water flowing through a garden hose. If the water keeps a consistent speed and pressure throughout its path, it is like saying the flow is incompressible. Understanding this helps us predict how quickly the hose will empty and how much water will come out at the end—essentially a practical application of the Navier-Stokes equations.
Stokes Hypothesis
Chapter 3 of 4
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However, if we want both of them to be same two different ways right. One is if you (()) (03:56) look at the above equation p bar will be = p either lambda + 2/3 mu = 0 or divergence of V = 0. So let us say =0 actually this is called Stokes Hypothesis all right. The second way is divergence of V can be if it is 0 then also this is possible. This is possible this is more commonly possible because divergence of V 0 for incompressible flow. So that is when we deal with water and hydraulics you see I mean all the time we assume incompressible flow.
Detailed Explanation
The Stokes Hypothesis is a theoretical condition used in fluid dynamics to simplify the relationship between mechanical and thermodynamic pressure in a fluid. According to this hypothesis, for the pressures to be equivalent, certain conditions must be met, such as the value of certain viscosity coefficients reaching specific thresholds (like lambda + 2/3 mu = 0). This doesn't often happen in practice, particularly because lambda is generally perceived as a positive quantity, making this hypothesis more of a mathematical convenience than a frequently occurring physical reality.
Examples & Analogies
Imagine you're trying to find the pressure of water flowing through a pipe. If you wanted the pressure of the water outside the pipe to match the pressure inside, you'd have to adjust the viscosity of the fluid perfectly—a tricky task! This is akin to balancing a scale perfectly. In real life, conditions like temperature and fluid composition always add variations, making exact matches rare, much like trying to find the perfect blend of ingredients in cooking.
Reduction to Euler's Equation
Chapter 4 of 4
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So actually we are going to study wave mechanics as part of Inviscid flow because there we will have potential theory and other things, but here in Inviscid the Euler it (()) (13:14) to discuss about mention Euler and the Bernoulli theorem how it has its origin okay. So if we assume that viscous terms are negligible as well in equation 21 then Navier–Stokes NS can be reduced to and this is equation number 22.
Detailed Explanation
In situations where the effects of viscosity can be ignored or are very small, the Navier-Stokes equations simplify to what are known as Euler's equations. These equations are particularly useful in describing inviscid flow—flow without viscosity. Understanding this transition from Navier-Stokes to Euler's equations helps engineers in various fields manage and predict fluid behaviors in scenarios like high-speed aerodynamics or wave interactions, where viscous forces have minimal impact.
Examples & Analogies
Consider the difference between swimming in a pool (viscous) versus moving through air (inviscid). In a pool, the water's viscosity slows you down, while in the air, you feel almost no resistance. Engineers use Euler's equations to model scenarios similar to swimming through air rather than water, making it easier to calculate how a plane moves through the atmosphere.
Key Concepts
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Navier-Stokes Equations: Fundamental equations governing fluid flow dynamics.
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Mechanical vs Thermodynamic Pressure: Distinction between stress-induced pressure and state-related pressure.
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Incompressible Flow: Scenario where fluid density is assumed constant, simplifying calculations.
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Stokes Hypothesis: Assumptions about stress conditions that lead to simpler fluid equations.
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Inviscid Flow: Describes fluid motion without accounting for viscosity.
Examples & Applications
The behavior of oil in a pipe versus water demonstrates varying fluid viscosities and how the Navier-Stokes equations apply.
Modeling pollutant dispersion in rivers using Navier-Stokes equations helps in environmental monitoring.
Memory Aids
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Rhymes
In flow fields dynamic and grand, / The Navier-Stokes equations make their stand.
Stories
Imagine a river with dense honey flowing through it, a metaphor for viscous fluid, where the struggle with its environment represents the forces in Navier-Stokes.
Memory Tools
Use 'N.S. - No Slip' to remember the no-slip condition in fluid mechanics.
Acronyms
D.O.V. - Deformation-Orientation-Velocity for understanding fluid strains.
Flash Cards
Glossary
- NavierStokes Equations
A set of equations that describe the motion of viscous fluid substances.
- Viscous Fluid
A fluid that has a significant resistance to flow, characterized by high viscosity.
- Thermodynamic Pressure
The pressure associated with the thermodynamic state of a fluid.
- Mechanical Pressure
Pressure that arises from internal viscous stresses within the fluid.
- Incompressible Flow
Flow in which fluid density remains constant.
- Inviscid Flow
Flow ignoring the effects of viscosity.
- Stokes Hypothesis
Assumption in fluid dynamics where certain stress conditions simplify fluid equations.
- Divergence of Velocity
A measure of how much the flow field spreads out from a point, important for fluid compressibility.
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